Residue Number System Based Building Blocks for Applications in Digital Signal Processing

Předkládaná disertační práce se zabývá návrhem základních bloků v systému zbytkových tříd pro zvýšení výkonu aplikací určených pro digitální zpracování signálů (DSP). Systém zbytkových tříd (RNS) je neváhová číselná soustava, jež umožňuje provádět paralelizovatelné, vysokorychlostní, bezpečné a proti chybám odolné aritmetické operace, které jsou zpracovávány bez přenosu mezi řády. Tyto vlastnosti jej činí značně perspektivním pro použití v DSP aplikacích náročných na výpočetní výkon a odolných proti chybám. Typický RNS systém se skládá ze tří hlavních částí: převodníku z binárního kódu do RNS, který počítá ekvivalent vstupních binárních hodnot v systému zbytkových tříd, dále jsou to paralelně řazené RNS aritmetické jednotky, které provádějí aritmetické operace s operandy již převedenými do RNS. Poslední část pak tvoří převodník z RNS do binárního kódu, který převádí výsledek zpět do výchozího binárního kódu. Hlavním cílem této disertační práce bylo navrhnout nové struktury základních bloků výše zmiňovaného systému zbytkových tříd, které mohou být využity v aplikacích DSP. Tato disertační práce předkládá zlepšení a návrhy nových struktur komponent RNS, simulaci a také ověření jejich funkčnosti prostřednictvím implementace v obvodech FPGA. Kromě návrhů nové struktury základních komponentů RNS je prezentován také podrobný výzkum různých sad modulů, který je srovnává a determinuje nejefektivnější sadu pro různé dynamické rozsahy. Dalším z klíčových přínosů disertační práce je objevení a ověření podmínky určující výběr optimální sady modulů, která umožňuje zvýšit výkonnost aplikací DSP. Dále byla navržena aplikace pro zpracování obrazu využívající RNS, která má vůči klasické binární implementanci nižší spotřebu a vyšší maximální pracovní frekvenci. V závěru práce byla vyhodnocena hlavní kritéria při rozhodování, zda je vhodnější pro danou aplikaci využít binární číselnou soustavu nebo RNS.This doctoral thesis deals with designing residue number system based building blocks to enhance the performance of digital signal processing applications. The residue number system (RNS) is a non-weighted number system that provides carry-free, parallel, high speed, secure and fault tolerant arithmetic operations. These features make it very attractive to be used in high-performance and fault tolerant digital signal processing (DSP) applications. A typical RNS system consists of three main components; the first one is the binary to residue converter that computes the RNS equivalent of the inputs represented in the binary number system. The second component in this system is parallel residue arithmetic units that perform arithmetic operations on the operands already represented in RNS. The last component is the residue to binary converter, which converts the outputs back into their binary representation. The main aim of this thesis was to propose novel structures of the basic components of this system in order to be later used as fundamental units in DSP applications. This thesis encloses improving and designing novel structures of these components, simulating and verifying their efficiency via FPGA implementation. In addition to suggesting novel structures of basic RNS components, a detailed study on different moduli sets that compares and determines the most efficient one for different dynamic range requirements is also presented. One of the main outcomes of this thesis is concluding and verifying the main condition that should be met when choosing a moduli set, in order to improve the timing performance of a DSP application. An RNS-based image processing application is also proposed. Its efficiency, in terms of timing performance and power consumption, is proved via comparing it with a binary-based one. Finally, the main considerations that should be taken into account when choosing to use the binary number system or RNS are also discussed in details.

The use of reversible logic gates in the design of residue number systems

Reversible computing is an emerging technique to achieve ultra-low-power circuits. Reversible arithmetic circuits allow for achieving energy-efficient high-performance computational systems. Residue number systems (RNS) provide parallel and fault-tolerant additions and multiplications without carry propagation between residue digits. The parallelism and fault-tolerance features of RNS can be leveraged to achieve high-performance reversible computing. This paper proposed RNS full reversible circuits, including forward converters, modular adders and multipliers, and reverse converters used for a class of RNS moduli sets with the composite form {2k, 2p-1}. Modulo 2n-1, 2n, and 2n+1 adders and multipliers were designed using reversible gates. Besides, reversible forward and reverse converters for the 3-moduli set {2n-1, 2n+k, 2n+1} have been designed. The proposed RNS-based reversible computing approach has been applied for consecutive multiplications with an improvement of above 15% in quantum cost after the twelfth iteration, and above 27% in quantum depth after the ninth iteration. The findings show that the use of the proposed RNS-based reversible computing in convolution results in a significant improvement in quantum depth in comparison to conventional methods based on weighted binary adders and multipliers

Accelerating DNN Training With Photonics: A Residue Number System-Based Design

Photonic computing is a compelling avenue for performing highly efficient matrix multiplication, a crucial operation in Deep Neural Networks (DNNs). While this method has shown great success in DNN inference, meeting the high precision demands of DNN training proves challenging due to the precision limitations imposed by costly data converters and the analog noise inherent in photonic hardware. This paper proposes Mirage, a photonic DNN training accelerator that overcomes the precision challenges in photonic hardware using the Residue Number System (RNS). RNS is a numeral system based on modular arithmetic\unicode{x2014}allowing us to perform high-precision operations via multiple low-precision modular operations. In this work, we present a novel micro-architecture and dataflow for an RNS-based photonic tensor core performing modular arithmetic in the analog domain. By combining RNS and photonics, Mirage provides high energy efficiency without compromising precision and can successfully train state-of-the-art DNNs achieving accuracy comparable to FP32 training. Our study shows that on average across several DNNs when compared to systolic arrays, Mirage achieves more than $23.8\times$ faster training and $32.1\times$ lower EDP in an iso-energy scenario and consumes $42.8\times$ lower power with comparable or better EDP in an iso-area scenario

Design and implementation of high-radix arithmetic systems based on the SDNR/RNS data representation

This project involved the design and implementation of high-radix arithmetic systems based on the hybrid SDNRIRNS data representation. Some real-time applications require a real-time arithmetic system. An SDNR/RNS arithmetic system provides parallel, real-time processing. The advantages and disadvantages of high-radix SDNR/RNS arithmetic, and the feasibility of implementing SDNR/RNS arithmetic systems in CMOS VLSI technology, were investigated in this project. A common methodological model, which included the stages of analysis, design, implementation, testing, and simulation, was followed. The combination of the SDNR and RNS transforms potential complex logic networks into simpler logic blocks. It was found that when constructing a SDNRIRNS adder, factors such as the radix, digit set, and moduli must be taken into account. There are many avenues still to explore. For example, implementing other arithmetic systems in the same CMOS VLSI technology used in this project and comparing them to equivalent SDNR/RNS systems would provide a set of benchmarks. These benchmarks would be useful in addressing issues relating to relative performance

Application of Residue Arithmetic in Communication and Signal Processing

Residue Number System (RNS) is a non-weighted number system. In RNS, the arithmetic operations are split into smaller parallel operations which are independent of each other. There is no carry propagation between these operations. Hence devices operating in this principle inherit property of high speed and low power consumption. But this property makes overflow detection is very difficult. Hence the moduli set is chosen such that there is no carry generated. In this thesis, the use of residue number system (RNS) is portrayed in designing solution to various applications of Communication and Signal Processing. RNS finds its application where integer arithmetic is authoritative process, since residue arithmetic operates efficiently on integers. New moduli set selection process, magnitude comparison routine and sign detection methods were limed on the onset of this dissertation. A good example of integer arithmetic is digital image. The pixels are represented by 8 bit unsigned number. Thus the operations are primarily unsigned and restricted to a small range. Hereby, in this thesis, a novel image encryption technique is depicted. The results show the robustness and timeliness of this technique. This technique is further compared to some of industry standard encryption algorithms for analysis based on robustness, encryption time and various other paradigms. Filters are signal conditioners. Each filter functions by accepting an input signal, blocking pre-specified frequency components, and passing the original signal minus those components to the output. A lowpass filter allows only low frequency signals (below some specified cutoff) through to its output, so it can be used to eliminate high frequencies. A novel design approach for a low pass filter based on residue arithmetic was also proposed. Some trite techniques as well as novel approaches were adopted to solve the design challenges. A technique for mapping the data in another space providing the liberty to work with floating numbers with a precision was adopted. PN sequence generator based on residue arithmetic is also formulated. This algorithm generates a pseudo-noise sequence which further was used to evince a spread spectrum multiuser communication system. The results are compared with trite techniques like Gold and Kasami sequence generators

Algorithmic Acceleration of B/FV-like Somewhat Homomorphic Encryption for Compute-Enabled RAM

Somewhat Homomorphic Encryption (SHE) allows arbitrary computation with nite multiplicative depths to be performed on encrypted data, but its overhead is high due to memory transfer incurred by large ciphertexts. Recent research has recognized the shortcomings of general-purpose computing for high-performance SHE, and has begun to pioneer the use of hardware-based SHE acceleration with hardware including FPGAs, GPUs, and Compute-Enabled RAM (CE-RAM). CERAM is well-suited for SHE, as it is not limited by the separation between memory and processing that bottlenecks other hardware. Further, CE-RAM does not move data between dierent processing elements. Recent research has shown the high eectiveness of CE-RAM for SHE as compared to highly-optimized CPU and FPGA implementations. However, algorithmic optimization for the implementation on CE-RAM is underexplored. In this work, we examine the eect of existing algorithmic optimizations upon a CE-RAM implementation of the B/FV scheme, and further introduce novel optimization techniques for the Full RNS Variant of B/FV. Our experiments show speedups of up to 784x for homomorphic multiplication, 143x for decryption, and 330x for encryption against a CPU implementation. We also compare our approach to similar work in CE-RAM, FPGA, and GPU acceleration, and note general improvement over existing work. In particular, for homomorphic multiplication we see speedups of 506.5x against CE-RAM, 66.85x against FPGA, and 30.8x against GPU as compared to existing work in hardware acceleration of B/FV

Measurements, Models, Systems and Design

531 s.